Question

a polynomial equation includes the term $8x^5$. according to the fundamental theorem of algebra, which statement must be true? (1 point)

all the roots of the equation are real.

the equation has an odd number of real roots.

the equation has more than 5 roots.

the equation has at least 5 roots.

Explanation:

Step1: Recall Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that a polynomial of degree \( n \) (where \( n \geq 1 \)) has exactly \( n \) roots (counting multiplicities) in the complex number system. For the given polynomial term \( 8x^5 \), the degree of the polynomial (at least) is 5. So the polynomial equation (of degree at least 5) will have at least 5 roots (when considering complex roots, and real roots are a subset of complex roots).

Step2: Analyze each option

• Option 1: "All the roots of the equation are real." A polynomial can have complex roots (e.g., \( x^2 + 1 = 0 \) has complex roots), so this is false.
• Option 2: "The equation has an odd number of real roots." The number of real roots can be even or odd (e.g., \( x^2 - 1 = 0 \) has 2 real roots, even), so this is not necessarily true.
• Option 3: "The equation has more than 5 roots." By Fundamental Theorem of Algebra, a degree \( n \) polynomial has at most \( n \) roots (counting multiplicities), so a degree 5 polynomial has at most 5 roots, so this is false.
• Option 4: "The equation has at least 5 roots." Since the polynomial has degree at least 5 (due to the \( 8x^5 \) term), by Fundamental Theorem of Algebra, it has exactly \( n \) roots (counting multiplicities) in complex numbers, so at least 5 roots (real or complex). This is true.

Answer:

The equation has at least 5 roots.